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m →2.3.5.7.19[53&147] (garibaldi nestoria): i dont remember why i wrote it as 53 & 147 when both x31eq and sintel's temp finder give it as 12&41 and when this is also the intuitive simple description that suggests 53 as a tuning
Cleanup; normalize mappings; review tunings and optimal ET sequences
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{{Wikipedia| Schisma }}
{{Wikipedia| Schisma }}


The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]). Tempering it out gives a [[5-limit]] microtemperament called [[Schismatic family#Schismatic aka Helmholtz|schismatic, schismic or Helmholtz]], which if extended to larger subgroups leads to the [[schismatic family]] of temperaments.
The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]).  


== Schismic temperaments derivable from its S-expressions ==
== Temperaments ==
Tempering out this comma gives a [[5-limit]] microtemperament called [[schismatic family #Schismatic aka helmholtz|schismatic, schismic or helmholtz]], which if extended to larger subgroups leads to the [[schismatic family]] of temperaments.


===[[Nestoria]]===
=== Nestoria ===
As the schisma is expressible as [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]), we can derive the 12&53 temperament:
{{See also| No-sevens subgroup temperaments #Nestoria }}


[[Subgroup]]: 2.3.5.19
As the schisma is expressible as [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]), we can derive the 12 & 53 temperament:


Patent EDO tunings: 12, 17, 24, 29, 36, 41, 53, 65, 77, 82, 89, 94, 101, 106, 118, 130, 135, 142, 147, 154, 159, 171, 183, 195, 207, 219, 248, 260, 272
=== Garibaldi ===
{{Main| Garibaldi }}


[[CTE]] generator: 701.684{{cent}}
===[[Garibaldi]]===
As the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]), we can derive the 41&53 temperament:
As the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]), we can derive the 41&53 temperament:


[[Subgroup]]: 2.3.5.7
==== 2.3.5.7.19 subgroup ====
 
{{Main| Garibaldi }}
Patent EDO tunings: 12, 29, 41, 53, 82, 94, 106, 135, 147
 
[[CTE]] generator: 702.059{{cent}}


==== 2.3.5.7.19[12&41] (garibaldi nestoria) ====
Adding Nestoria to Garibaldi (tempering [[400/399|S20]]) results in an extremely elegant temperament which has all of the same patent tunings that Garibaldi has but which includes a mapping for 19 through Nestoria.
Adding Nestoria to Garibaldi (tempering [[400/399|S20]]) results in an extremely elegant temperament which has all of the same patent tunings that Garibaldi has but which includes a mapping for 19 through Nestoria.


[[Subgroup]]: 2.3.5.7.19
=== 2.3.5.7.17 12 & 118 & 171 (unnamed) ===
As the schisma also equals [[57375/57344|S15/S16]] * [[1701/1700|S18/S20]], we can derive the extremely accurate 12&41 temperament:


Patent EDO tunings: 12, 29, 41, 53, 82, 94, 106, 135, 147
[[Subgroup]]: 2.3.5.7.17


{{mapping|legend=1| 1 1 7 11 6 | 0 1 -8 -14 -3 }}
[[Comma list]]: 1701/1700, 32805/32768


[[CTE]] generator: 702.043{{cent}}
{{mapping|legend=1| 1 0 15 0 -32 | 0 1 -8 0 21 | 0 0 0 1 1 }}


=== 2.3.5.7.17[12&130&171] (unnamed) ===
: mapping generators: ~2, ~3, ~7
As the schisma also equals [[57375/57344|S15/S16]] * [[1701/1700|S18/S20]], we can derive the extremely accurate 12&41 temperament:


[[Subgroup]]: 2.3.5.7.17
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7197, ~7/4 = 968.8307


Patent EDO tunings < 300 (largest is 2548): 12, 29, 41, 53, 118, 130, 142, 159, 171, 183, 212, 224, 236, 289
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 472, 525, 643, 814, 985, 1799, 2324, 2495, 3138b, 3309bd, 4294bdg }}


{{mapping|legend=1| 1 1 7 2 -9 | 0 1 -8 0 21 | 0 0 0 1 1 }}
==== 2.3.5.7.17.19 12 & 118 & 171 (unnamed) ====
By tempering [[1216/1215|S16/S18]] we equate [[225/224|S15]] with [[400/399|S20]] (tempering the other comma of Nestoria) because of S15~S16~S18~S20, leading to:


[[CTE]] generators: (2/1,) 3/2 = 701.72{{cent}}, 7/4 = 968.831{{cent}}
[[Subgroup]]: 2.3.5.7.17.19


==== 2.3.5.7.17.19[12&130&171] (unnamed Nestoria) ====
[[Comma list]]: 361/360, 513/512, 1701/1700
By tempering [[1216/1215|S16/S18]] we equate [[225/224|S15]] with [[400/399|S20]] (tempering the other comma of Nestoria) because of S15~S16~S18~S20, leading to:


[[Subgroup]]: 2.3.5.7.17.19
{{mapping|legend=1| 1 0 15 0 -32 9 | 0 1 -8 0 21 -3 | 0 0 0 1 1 0 }}


Patent EDO tunings: 12, 29, 41, 53, 118, 130, 142, 159, 171, 183
: mapping generators: ~2, ~3, ~7


{{mapping|legend=1| 1 1 7 2 -9 6 | 0 1 -8 0 21 -3 | 0 0 0 1 1 0 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7053, ~7/4 = 968.9281


[[CTE]] generators: (2/1,) 3/2 = 701.705{{cent}}, 7/4 = 968.928{{cent}}
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }}


== Trivia ==
== Trivia ==
Retrieved from "https://en.xen.wiki/w/Schisma"